# Talks

## Finding Holes: Homology and Cohomology

20th September, 2021

This is a talk I gave a talk for the Malta Mathematical Society as part of the Differential Geometry Lecture Series. First I gave an informal introduction to the ideas of homology and cohomology, as two different strategies for identifying holes in manifolds. Subsequently, I described de Rham's theorem, showing that homology and cohomology measure the same "holes", and Stoke's theorem, the higher dimensional analogue to the fundamental theorem of calculus.

## Eigenvalues in Number Theory: Ramanujan Graphs

29th April, 2021

29th April is "Eigenvalue Day", and so the Malta Mathematical Society organised a series of talks on eigenvalues. For the occasion, I gave a talk on a use of eigenvalues in number theory—in particular, I described the use of the second largest eigenvalue as a measure of connectivity in a graph, the idea of Ramanujan graphs, and why they make good expanders.

## Elementary Number Theory: Quadratic Reciprocity

22nd April, 2021

Quadratic Reciprocity is one of the most important theorems in all of mathematics. In this talk for the Malta Mathematical Society, I motivated quadratic reciprocity by introducing modular arithmetic and finite fields. I subsequently computed some Legendre symbols using the result.

## Factorials of Large Numbers

18th November, 2020

In this talk, I introduced the idea underlying the technique of partial summation which is often applied in analytic number theoryto obtain asymptotic bounds for sums by expressing them in terms of integrals. Using this technique and its more general form (the Euler–Maclaurin formula), one can obtain Stirling's approximation for factorials. I gave an estimate of 10000! using this approximation; and ended by discussing the de Polignac formula for factors of N!.

## Smooth Submanifolds

6th July, 2020

As part of a Differential Geometry reading group at the University of Warwick Mathematics Institute, I gave a talk on chapter 5 of Lee's book on Smooth Manifolds, about smooth submanifolds. The talk comprises embedded submanifolds, immersed submanifolds, and the tangent space to a submanifold.

## The Lambda Calculus

18th December, 2019

I gave a lecture on Church's \(\lambda\)-calculus (Lambda calculus). The \(\lambda\)-calculus is a very simple formal language which is expressive enough to serve as a foundation for mathematical logic. In the talk, I discussed what a formal language is, explored some of the history of mathematical logic and finally demonstrated how the \(\lambda\)-calculus can model various mathematical structures. The \(\bf Y\)-combinator was also mentioned briefly.

## Synthesising Runtime Enforcement Monitors for \(\boldsymbol\mu\textbf{HML}\)

10th June, 2019

This talk is on my undergraduate computer science dissertation. I considered a subset \(\text{sHML}\) of formulæ in the Hennessy-Milner Logic with recursion (\(\mu\text{HML}\)) which are enforceable through suppressions. A synthesis function is introduced, which converts safety properties in \(\text{sHML}\) to suppression enforcers through a formula normalisation process.

## Walks and CDCs of Comain Graphs

12th April, 2019

The S³ Science Conference is an event organised by the Faculty of Science students' society, as an opportunity for students of the faculty to present their work to their fellow students. In 2019, the event was held at Esplora, a science museum in the south of Malta. I presented research related to a publication I coauthored with my undergraduate mathematics dissertation supervisor, Prof. Irene Sciriha.

## Two-Graphs and Conference Graphs

31st January, 2019

The Combinatorics and Graph Theory Day was an event organised by the Department of Mathematics at the University of Malta, where research was presented by researchers from the department, as well as researchers from the Mathematics Department of the University of Pannoia, Hungary. I presented research I conducted on two-graphs and conference graphs as an undergraduate student.

## Two-Graphs and NSSDs: An Algebraic Approach

2nd March, 2018

The S³ Science Conference is an event organised by the Faculty of Science students' society, as an opportunity for students of the faculty to present their work to their fellow students. I presented research I conducted on two-graphs and non-sigular graphs with a singular deck (NSSDs).